# Typical properties entanglement on manifolds of isospectral density matrices

Presenter: Michal Oszmaniec (CFT PAS)

Co-author: Marek Kus (CFT PAS)

### Abstract

We analyze typical properties of entanglement on manifolds of isospectral density matrices. Considered systems include arbitrary finite number of distinguishable, bosonic or fermionic particles. In the case of distinguishable particles and bosons we adopt the usual definition of entanglement. In the case of fermions we define a given state to be “entangled” if it cannot be expressed as a convex combination of projections onto Slater determinants. We start from the uniform measure on the space of isospectral density matrices and ask a natural question: what fraction of mixed states with the fixed spectrum are “entangled”. We derive new concentration inequalities that generalize the well-known Levy lemma to arbitrary manifold of isospectral density matrices. We apply these inequalities to functions that are lower bounds for of the generalized quantum concurrence, $C\left(\rho\right)$ - a non-negative function defined on the state of all mixed states vanishing precisely for non-entangled mixed states. Application of concentration inequalities to lower bounds for the concurrence allows us, provided that the purity of the state is large enough, to estimate from below the fraction of “entangled” states with a given spectrum. By considering the “dual” upper bounds and repeating the same concentration arguments, we obtain, again under the condition that the purity is sufficiently large, the lower bounds for the fraction of states for which upper and lower bounds of the concurrence are close to their actual values.